M II I Ml III I II II I II 

019 423 818 6 



Hollinger Corp. 
P H 8.5 



tJ-Jl 



WILLIAMS- 



UNIVERSAL TRAVERSE TABLE, 



WITH 



EXPLANATIONS AND ILLUSTRATIONS; 



BY THE 



AID OP WHICH ANY PERSON ACQUAINTED WITH COMMON DECIMALS, MAY 

CALCULATE HEIGHTS, DISTANCES, ALTITUDES, ELEVATIONS, 

AND EIGHT-ANGLED TRIANGLES GENERALLY. 



BY JNO. S. WILLIAMS. 




STEREOTYPED BY J. A. JAMES CINCINNATI. 



(Etuctnuatf: 

PUBLISHED BY U. P. JAMES, AND 
HUBBARD & EDMANDS. 

1833. 



J. A. JAMES, PRINTER CINCINNATI. 



l/i;20 



sf/f 






HOS 



WILLIAMS'S UNIVERSAL TRAVERSE TABLE, WITH EX- 
PLANATIONS AND ILLUSTRATIONS; 

By the aid of which any person, acquainted with common deci- 
mals, may calculate heights distances, altitudes, elevations, and 
right angled triangles generally. 



Even degrees. 



Latitude.) Depart. 



1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 



.999849 
999391 
.998627 
.997564 
996199 
.994520 
.992543 
.990268 
987689 
984807 
981627 
978148 
974369 
970296 
965927 
.961262 
.956304 
.951054 
.945517 
.939693 
.933560 
.927183 
.920504 
.913544 
.906308 
.898773 
.891006 
.882947 
.874650 
.866026 
.857171 
.848047 
.838670 
.829037 
.819153 
.809017 
.798637 
.788010 
.777146 
.766044 
.754710 
.743145 
.731351 
.719340 
,.707107 



Depart. 



.017451 

034899 

.052336 

.069757 

.087156 

.104529 

.121861 

,139173 

.156434 

,173648 

,190809 

.207912 

,224951 

,241922 

.258819 

.275637 

-292372 

.309017 

.325568 

.342020 

.358368 

.37460tf 

.390733 

.406736 

.422618 

.438366 

.453991 

.469471 

.484809 

.500000 

.515038 

.529920 

.544639 

.559193 

.573576 

.587784 

.601815 

.615661 

.629320 

.642787 

.656959 

.669131 

.681997 

.694657 

.707107 



89 
88 
87 
86 
85 
84 
83 
82 
81 
80 
79 
78 
77 
76 
75 
74 
73 
72 
71 
70 
69 
68 
67 
66 
65 
64 
63 
62 
61 
60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 



Lat. 



Even degrees. 



O I One quarter. | One half. | Three quarters. I 



|Latitude.| Depart, j Latitude.! Depart. jLatitude. | Depart. 



,999997 

999763 

.999230 

,998399 

,997250 

,995805 

,994056 

■992004 

,989650 

,986998 

,984004 

,980786 

,977231 

.973378 

.969231 

.964789 

.960051 

.955020 

.949700 

.944087 

.933191 

.932009 

.925540 

.918770 

.911763 

.904458 

.896875 

.889016 

.880892 

.872495 

.863836 

.854912 

.845730 

.836287 

.826589 

.816642 

.806444 

.796002 

.785316 

.774393 

.763233 

.751840 

.740219 

.728372 

.716302 



004363 
021815 
039260 
056693 
074108 
.091508 
.108867 
126199 
.143493 
.160742 
177943 
195091 
212178 
.229200 
.246152 
.263030 
.279829 
.296542 
.313164 
.329700 
.346117 
.362438 
.378648 
.394744 
.410719 
.426569 
.442289 
.457874 
.473320 
.488621 
.503775 
.518771 
.533615 
.548294 
.562805 
.577145 
.591310 
.605293 
.619086 
.632704 
.646124 
.659345 
.672366 
.685184 
.697790 



.999960 
.999658 
.999050 
.998134 
.996916 
.995395 
.993566 
.991445 
.989018 
.986286 
.983255 
.979925 
.976298 
.972371 
.968149 
.963631 
.958820 
.953717 
.948324 
.942649 
.936674 
.930417 
.923879 
.917062 
.909960 
.902585 
.894932 
.887010 
.878818 
.870356 
.861628 
.852641 
.843392 
.833887 
.824107 
.814115 
.803857 
.793355 
.782607 
.771625 
.760407 
.748955 
.737278 
.725373 
.713251 



Depart. | Lat. ( Depart. 



.008727 
.026177 
.043620 
.061048 
.078459 
.095846 
.113204 
.130526 
.147809 
.165048 
,182236 
.199368 
.216439 
.233445 
.250380 
.267239 
.284016 
.300748 
.317304 
.333807 
.350207 
.366501 
.382683 
.398750 
.414693 
.430511 
.446198 
.461749 
.477160 
.492424 
.507538 
.522499 
.537299 
.551937 
.566406 
.580703 
.594832 
.608762 
.622516 
.636079 
.649448 
.662621 
.675591 
.688354 
.700910 



Lat. 



.999914 

.999532 

,998848 

.997859 

.996561 

.994968 

.993068 

,990866 

.988360 

.985555 

982452 

979045 

975342 

971342 

966971 

.962456 

.957571 

.952393 

.946930 

.941178 

.935135 

.928811 

.922202 

•915313 

.908142 

•900698 

•892978 

.884967 

.876726 

•868196 

•859408 

.850353 

.841037 

.831470 

•821647 

•811572 

•801254 

.790689 

.779884 

.768842 

.757565 

.746057 

.734322 

.722363 

.710185 



013082 
.030538 
047972 
065403 
082808 
.100188 
.117510 
134852 
152124 
169347 
186524 
203642 
220656 
237680 
254884 
.271440 
.288196 
.304864 
.321440 
.337917 
.354290 
.370558 
.386711 
.402746 
.418660 
.434445 
.450089 
.465615 
.480988 
.496216 
.511293 
.526213 
.540955 
.555571 
.569996 
.584249 
.598325 
.612218 
.625923 
.639438 
.652758 
.665882 
.678800 
.691513 
.703990 



89 

86 

87 

86 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

61 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 



Depart.j Lat. 



three quarters. 



one half. j one quarter. 



Explanation.— If the degrees sought are under 45, they stand on the left; 
but if over 45, they stand on the right of the columns of latitude and depar- 
ture. The terms "latitude and departure," and the quarter degrees at the top 
belong to the degrees under 45, found on the left; but the terms "departure 
and latitude," and the quarters at the bottom belong to the > degrees over 45, 
found on the right; thus: the latitude of 15* degrees, is .964789; and its depar- 
ture .263030, and the departure of 74| is .964789, and its latitude .263030. 



Entered according to act of Congress, in the year 1833 by John S. Williams, in the Clerk'. Office of the District 

of Ohio. . 



3 

EXPLANATIONS AND ILLUSTRATIONS. 

It is not pretended that this traverse table presents in all cases, an 
easier or better method of calculating triangles than is known to 
mathematicians ; but the intention is, to place before those arithmeti- 
cians, who have not the opportunity of studying the higher branches 
of science, the means by which they can avail themselves of the 
practical use of triangles. It is however believed that for the solution 
of many problems solvable by the use of this table,, that no mathema- 
tician knows of a more easy, speedy, or accurate plan of ascertaining 
the value of lines and angles. * 

By a little attention on the part of the ingenious arithmetician to the 
principles here laid down and illustrated, he will acquire much benefi- 
cial knowledge of geometry. The problems solved comprehend most 
of the principles of triangles. There are few cases useful in practice 
that cannot be wrought out by common decimal fractions, aided by the 
universal traverse table. It must be especially borne in mind that the 
latitude and departure given are decimals, and in using them, whether 
it be to add, subtract, multiply, divide, or reduce, the same rules must 
be observed in respect to pointing off, &c. as are laid down in the 
arithmetics for working by common decimals. 

Geometricians suppose circles great and small to be divided into 360 
equal parts called degrees, and each degree divided into 60 equal parts 
called minutes. It is evident then that the larger the circle that is thus 
divided, the longer will the degrees and minutes be. The degree is 
the one three hundred and sixtieth part of the circumference, let that 
circumference be long or short. In a circle large enough just to 
encompass our earth, each degree is 69i miles long, but in a circle as 
small as the dotted line in figure 1, the degrees are very short. 
Geometricians also suppose that the point at which any two straight 
lines meet, as at A. may be the centre of the circle upon which the 
degrees between the two lines may be counted. It is perfectly imma- 
terial whether the circle is described in the middle or at the end of the 
straight lines. In each and every one the degrees between them 
would be the same in number though not in length. The opening 
between the lines is called an angle, and the size of the angle is 
counted in degrees upon the circle. If the number of degrees in an 
angle be exactly 90 or one quarter of the circle, the angle is called a 
right angle. A greater angle is called an obtuse angle, a less one 
acute. 

When three straight lines meet so as to enclose a space, that space 
is called a triangle, because it has three angles or corners as A. B. C. 
in the figure. If one of those angles be a right angle as at C. the 
triangle is called a right angled triangle. 

O DO© 

Take three staves of equal length, whether that length be one foot, 
one yard, one pole, or any other length whatever, set one staff upon 
another as C. B. upon A. C. so as to form a right angle or square at C. 
Take the other staff A. B. and place it even with the end of A. Cat 
A. leaning it against the upright one at B. and you have formed a 
right angled triangle A. B. C. with the right angle at C. Then if pre- 
viously the staves A. C. and C. B. should have been divided into 10. 




Latitude 

qiiiiliii! ] iii] | ii i i ; i ii i|iii; | i i i i| iiii | iTii [M ii | i iin i i ii |mi |iii p n| r i i i | iii i |ii ir( > 

70 80 90~W0 



10 



20 30 40 50 60 



100, 1000, or any num- 
ber of equal decimal 
parts, you have scales 
upon which to reckon 
the parts of all right 
angled triangles form 
ed of the same staves. 
The staff or line A. B. 
is called one, so is the 
whole length of each of 
the staves. The num- 
ber of parts from A. to 
C. when forming the tri- 
angle is called the lati- 
tude, and the number of 
parts from C. to B. is 
called the departure of 
the line A. B. In the 
table these two staves 
or any staves or lines so 
situated, are presumed 
to be divided into one million of parts each, being the decimal parts of 
the integer A. B. The angle or opening between the staves A. B. and 
A. C. as counted on the circle in the figure is 37 degress, accordingly 
as you will find the latitude and departure opposite to 37 degrees in 
the tables, you find that C. B stands upon A. C. very nearly at the 80th 
division, and that A. B. touches C. B. about the 60th division, repre- 
senting the latitude and departure of A. B. Set the staff C. B. any 
where upon the line or staff A. C. and lean the line or staff A. B. 
against it as before, and the intersection at B. will always cut the 
circle at the degree represented at the sides of the table, while the 
latitude and departure opposite to the degree in the table, will corres- 
pond with the divisions on the lines A. C. and C. B. as marked by the 
points C. and B, 

In all right angled triangles it is known that the two angles at A. 
and B. added together make just 90 degrees. That as one is increased 
the other is diminished. In the example before us the angle at B. is 
53 degrees, that being the complement of 37 degrees or what 37 
degrees lacks of being 90 degrees. Make the angle at A. 25£ degrees 
and that at B. will be 64£, and so of any number of degrees and parts 
under 90. In this case A. B. would touch C. B. at .43 -|- which is the 
latitude of 64£ and departure of 25i. and C. B. would stand upon 
A. C. at .90 -\- the departure of 64i and the latitude of 25i as shown 
in the table. It is thus that the latitude of all angles under 45 degrees 
is the departure of all angles over 45 degrees, of which such angles 
are the complements. Hence one half of the figures and size of a 
traverse table is saved by reversing the terms latitude and departure 
at top and bottom as applied to angles under and over 45 degrees. 

It is known that as long as the angle at A. remains the same, whether 
it be 37 or any other number of degrees, or degrees and parts, that the 
other parts of the triangle will bear the same proportion to the line 



or unit A. B., let the triangle be enlarged or diminished ever so much. 
Thus if A. B. is made 3, 4, 20, or any number of times as long as that 
in the figure, the latitude and departure or the lengths A. C. and 
C. B. will be 3, 4, 20, or any other [the same] number of times the 
lengths that they are in the figure. 

In using the table, unless' great accuracy is required, or the lines 
very long, it is unnecessary to use more than three, four or five of the 
figures. The table is carried so far into decimal parts that if A. B. 
were one mile long, one in the table would be about one sixteenth 
of an inch. There may be as many rejected from the right hand as the 
nature of the case will admit, taking care to increase your right hand 
figure one, if the left one, you reject be 5 or over. Thus the departure 
of 31 degrees might be set down .515 — that of 37 degrees .602 and 
that of 32 degrees .530 or .53. Should you wish to know the latitude 
and departure answering to any number of minutes less than 15 or a 
quarter of a degree, take out the latitude for the quarter next greater, 
and next less than the deg. and minutes sought. Subtract the less from 
the greater latitude; multiply the difference by the number of odd 
minutes, divide the product by 15 and subtract the quotient from the 
greater latitude taken out, and the remainder is the latitude sought 
For finding the departure the process is the same, except the quotient 
found is to be added to the less departure taken out; the sum is the 
departure of the degree and minutes. 

What is the latitude and departure of 42 degrees and 17 minutes? 
17 minutes being one quarter and two minutes, lat. 42^ .740219 less 
by lat. of 42i .737278 = 3021 X 2 and divided by 15 = 403. .740219 
— 403 = 739816 the lat. of 42 degrees and 17 minutes. Again dep. 
of 42i .675591 —dep. of 42* .672366 = 3225 X 2 and divided by 
15 = 430. .672366 + 430 = .672796 the departure of 42 degrees 
and 17 minutes. 

APPLICATION. 

What is the rise of 10 feet at 3 degrees of elevation? Dep. of 3 deg. 
= .052336 X 10 = .52336 feet X 12 = 6.28032 inches, that is 6 
inches and a little over a quarter, answer. I find a plane descends 7 
inches in 33 feet, what is the degree of descent? Divide 7 inches by 12 
gives .583333 of a foot. Divide this by 33 feet gives .017676 being 
the departure of one foot at the rate of 7 inches in 33 feet. In the 
table is found that the departure of 1 foot at 1 degree is 017451 so 
that one degree is the descent of the plane very nearly. I find from 
the centre to the eave of a house is 17 feet and that the face of the 
roof from eave to comb is 21 feet, what degree of slope has the roof? 
Divide 17 feet which is A. C. in fig. 1 by 21 which is A. B. and you 
have .809524 as the proportional latitude and by searching the table 
for the nearest latitude you find that the slope of the roof to be a little 
less than 38 degrees. From the centre of a house to the eaves on 
each side is 20 feet. It is desirable to pitch the roof 40 degrees; 
how high must the comb be raised above the square and how much 
will the roof face on each side? As .766 latitude of 40 deg. is 
to .643 dep. so 20 ft. is to 16.78 ft rise answer. 20 feet divided by lat. of 
40 deg. .642787 = 26.1 feet answer face of roof or A.B. in fig. 1. 



i There is a church with a spire erected over its centre, my eye is 
480 feet from the centre of the church and level with the floor; I find 
the elevation of the spire to be 20 degrees, query its height above the 
floor, and its distance from my eye? As .9397 the lat. of 20 deg. is 
to .342 the dep. of 20 deg. so 480 feet is to 174.7 feet height of spire. 
Divide 480 feet by .9397 lat. of 20 deg. gives 510.8 feet the distance of 
the spire from my eye or A. B. of the figure. I change my position on 
level ground and now find the spire elevated 23i degrees, what is my 
present distance from the centre of the church? As dep. of 23i deg. 
.3987 is to lat. of 23i degrees .917, so 174.7 feet is to 401.8 feet, my 
present distance from the centre of the church. 

rises perpen- Figure 2 ^y^ tWBP^ 

shore, at C. the "5^"1j54 \ ^/W** (fllilllil 

degrees from 

my eye, which is ten feet above the water. I recede from the shoro 
on level ground 72 feet, and find the top of the cliff elevated from this 
station, at D., 25£ degrees : What is the distance of my eye at each 
station from the top of the cliff, also the perpendicular height of the cliff 
above the water and the breadth of the stream, C. B.? 36^ — 25i = 
10| degrees, the angle D. A. C. ; for as one of the acute angles of a right- 
angled triangle is diminished the other is increased, as before stated: 
D. A. B. and C. A. B., being rightangled triangles, having the common 
perpendicular A. B. 

72 X -430511, the dep. of 251 = 30.996792= C. a. 
72 X .902589, the lat. of 25£ =64.98612= D. a. 
As .186542, dep. of 10f degs. is to .982452, lat. of lOf degs: so 30. 
.996792. = C. a. is to 163.2651 feet = A. a. A. a. 163.265 + D. a. 
64.986 = D. A. 228.251; which is the distance from the 2d station D. 
to top of precipice A. 

C. a. 30.996792 divided by .186524 dep. of 10f degs. = 166.181 feet 
C. A. ; the distance from 1st station C. to top of cliff. Line D. A. 228.251 
X .430511 dep. of 25* degs. = 98.2645 feet; the height of the cliff 
above the level of the eye ; add 10 = 108.264 feet, the height of the top 
of cliff from water. 

Line C. A. 166.181 X .591310 dep. of 36* degs. = 98.264 feet; the 
height of top of cliff above the level of the eye at 1 st station, add 10 feet 
the height of eye above water, gives 108.264 feet, the height of cliff; 
the same as before. 

166.181 C. A. X .806444 lat. of 364 = 134.016 feet, width of the 

228.251 D. A X .902585 lat. of 25i == 206.016, from which take 72 



feet, distance between stations, = 134.016 feet, the width of stream, 
and same as before. 

By the same principles of solution, horizontal angles, taken with a 
compass, may be wrought as well as most if not all the most beneficial 
and practical values of right lines and angles may be determined with- 
out recourse to trigonometry. 

The mechanical power of inclined planes is inversely proportional to 
their departures very nearly, if the inclination be not great. At 8 de- 
grees of inclination, the error amounts to less than .01 of the power. 
At 25 degrees, it amounts to less than .1 of the power, being equal to 
the excess of 1. over the latitude of the plane. 

What is the angle at the point of a wedge, the power of which will be 
25 fold, that is to raise 25 pounds for every pound of power applied 
over and above the resistance of friction? 1. divided by 25. =.04; which 
is a little more than the departure of 2\ degrees — which is the answer, 
very nearly. 

I find the thread of a screw forms an angle of three degrees with a 
rule laid perpendicular to or directly across the spindle, — what is its 
power? 1. divided by .052336, the dep. of 3 degrees, = 19 fold; which 
is the power of the screw. 

It is found, that the power necessary to move a common wagon on 
a level smooth road, is equal to _Lth of its weight. At what degree of 

inclination must such a road he made, so that a wagon will just stand or 
run upon it, that is, upon which the inclination of the wagon to descend 
will just equal the friction? 1. divided by 29 = .034483; which is the 
dep. of 2 degrees very nearly, or the inclination of road required. 

What portion of the weight of a carriage and load pushes against the 
wheel horses in descending such a road at 4£ degrees of inclination, 
with the wheels unlocked? .078459 dep. of 4i degs., lessened by the 
departure to balance friction above .034483 == .043976. 1. divided by 
this sum gives 22.7, nearly; or about 1 pound for every 22.7 pounds of 
weight ; or ninety nine pounds to the ton of wagon and load, nearly. 

What is the force required to draw carriages up the same plane? 
.078459 dep. of 4i degs. increased by the above sum .034483 for fric- 
tion, gives .112942. 1. divided by this sum gives 8.8: that is, one 
pound of force for each 8.8 pounds of wagon and load; or about 255 
pounds to the ton of wagon and load. 

I find that J_th of the weight of a horse applied at right angles to the 

arms of a tread-wheel, will produce the requisite power: What inclina- 
tion must be given to the wheel, so that the horse walking upon it will 
give the necessary power? 1. divided by 7. gives .142857, or a trifle 
less than the departure of 8i degrees, which is the inclination required. 

A COMPREHENSIVE REDUCTION TABLE. 
LONG MEASURE. 





Inches. 


Feet. 


Yards. 


Poles. 


1 league is 


190,080. 


15,840. 


5,280. 


960. 


1 miie is 


63,360. 


5,280. 


1,760. 


320. 


1 chain is 


792. 


66. 


22. 


4. 


1 pole is 


198. 


16.5 


5.5 


1. 


1 yard is 


36. 


3. 


1. 


.1818184- 


1 foot is 


12. 


1. 


.03334- 


.060606 r 


1 inch is 


1. 


.0834- 


.0277-f- 


.005051— 



Chains. 


Miles. 


240. 


3. 


80. 


L 


1. 


.125 


55 


.003125 


.045455— 


.000568-f 
.000169 f 


.015152— 


.001263— 


.000016— 



LIBRARY OF CONGRESS 4 






. 019 423 818 6 



019 423 818 6 f 



Hollinger Corp. 
pH 8.5 



LIBRARY OF CONGRESS 

II 



019 423 818 6 4 




Hollinger Corp. 
pH 8.5 



